ProbabilityPart_3.CardsBagsAndOdds

ProbabilityPart_3.CardsBagsAndOdds - 1 Added notes on...

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1 Added notes on conditional probability. These are further notes on conditional probability. Recall that we have in mind two decks of cards, the ordinary deck that contains 52 cards, and what we call the castaway s deck which contains only 31 cards. When we finished class last time we were thinking about the question that I am going to write out very carefully here: 1.1 Two bags of cards Suppose I pick one of the two decks randomly. I can do this by putting all the cards from one deck in one black bag and putting all the cards from the other deck in another black bag mixing up the bags and then randomly picking one of the bags. So then I don’t know which deck I’ve got . And then I reach into the deck and I pull out a red jack. Of course if I look at the ca rd I actually know whether it’s the jack of hearts or the jack of diamonds but this is a mathematics class so we are going to say that I don’t know that . I just know that I picked out a red jack. Now the question is, does knowing that I’ve picked the re d jack help me to guess whether I have picked from the full deck or from the castaways deck. When the class was ending as I recall it, several of you suggested that it probably meant that I was somehow more likely to have chosen from the castaway s deck, but we didn’t really make the idea completely clear. What we’re going to try to do now is work out the idea very clearly. I’m going to begin by pointing out that the conditions of the experiment are very , very important. The condition was that I picked a deck and then after that I picked the card. 1.2 One Bag of cards Now, suppose I have not done that. Suppose I had taken all 52 cards from the good deck, and all 31 cards from the castaway s deck and I had dropped them all into one sack. We can imagine that after we get the cards out we can tell which deck they came from because the cards in the castaway deck are all kind of waterlogged, since they were rescued when a boat sank. Under this new experiment suppose I tell you that I’ve picked a red jack. And I ask you which is more likely, is it the red jack from the full deck, or the red jack from the castaways deck?
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Well, you should reason this way. There are in fact 4 red jacks in the bag. And the fact that I am picking randomly means that any one of them has an equal chance to be picked.
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